Turaev [defined a simple-homotopy invariant][1] which is a complete invariant of homeomorphism type (originally assuming geometrization). 

Here is the Springer link if you have a subscription:
[Towards the topological classification of geometric 3-manifolds][2]

He claims in the paper that a map between closed 3-manifolds is a
homotopy equivalence if and only if it is a simple homotopy equivalence,
but he says that the proof of this result will appear in a later paper. I'm 
not sure if this has appeared though (I haven't searched through his
later papers on torsion, and there's no MathScinet link). 

  [1]: http://www.ams.org/mathscinet-getitem?mr=970081
  [2]: http://www.springerlink.com/content/q3752914430453q7/